Since our fits include observables which are sensitive to the MPS at weakly non-linear scales one might wonder about the importance of non-linear effects. To probe the sensitivity of our results to non-linear physics we evaluated the \(\chi ^2\)  values of several of our best fit points with and without using halofit [1]}, [2]}. We found slight improvements in the fits to the CMB and CMB lensing (with overall \(\delta \chi ^2\sim 2-4\) ) with no significant differences in the level of improvement between models. We conclude that the impact of non-linear effects on our analysis is small and we only present results without halofit. We also note that \(S_8\)  is defined as an integral over the linear matter power spectrum which means that its sensitivity to non-linear physics is a subtlety that must be faced in the extraction of \(S_8\)  from observations, not in the theory calculation. Since halofit is tuned to reproduce the non-linear effects for \(\Lambda {\rm CDM}\)  it is not necessarily accurate for WZDR+ and a full analysis including non-linear effects in WZDR+ is beyond the scope of this work.
